Topology of tame contractible $n$-manifolds, $n=6,7$, and BiRicci Curvature

Abstract

We show that if $M^n$, $n=6,7$, is the interior of a compact, contractible $n$-manifold with boundary $X$, such that $π_i(X,\partial X)=0$, $3\leq i \leq n-3$, and supports complete metrics with positive BiRicci curvature with $C$-quadratic decay at infinity for some $C>\frac{n^2}{4}$, then $M$ is diffeomorphic to $\mathbb{R}^n$. Furthermore, we construct a compact, contractible Newman $n$-manifold $N^n$, $n\geq6$, based on a presentation of the Higman group such that $π_i(N,\partial N)=0$, $3\leq i \leq n-3$, and $\mathrm{int}(N)$ admits a metric with uniform positive scalar curvature.